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Theorem cnvss 4670
 Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
cnvss (𝐴𝐵𝐴𝐵)

Proof of Theorem cnvss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3055 . . . 4 (𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2 df-br 3894 . . . 4 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3 df-br 3894 . . . 4 (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
41, 2, 33imtr4g 204 . . 3 (𝐴𝐵 → (𝑦𝐴𝑥𝑦𝐵𝑥))
54ssopab2dv 4158 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6 df-cnv 4505 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7 df-cnv 4505 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
85, 6, 73sstr4g 3104 1 (𝐴𝐵𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1461   ⊆ wss 3035  ⟨cop 3494   class class class wbr 3893  {copab 3946  ◡ccnv 4496 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095 This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-in 3041  df-ss 3048  df-br 3894  df-opab 3948  df-cnv 4505 This theorem is referenced by:  cnveq  4671  rnss  4727  relcnvtr  5014  funss  5098  funcnvuni  5148  funres11  5151  funcnvres  5152  foimacnv  5339  tposss  6095  structcnvcnv  11812
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